a comparative study of the harmonic and arithmetic

The INL is a U.S. Department of Energy National Laboratory operated by Battelle Energy Alliance

INL/EXT-08-13999

A Comparative Study of the Harmonic and Arithmetic Averaging of Diffusion Coefficients for Non-Linear Heat Conduction Problems

Samet Y. Kadioglu Robert R. Nourgaliev Vincent A. Mousseau

March 2008

INL/EXT-08-13999

A Comparative Study of the Harmonic and Arithmetic Averaging of Diffusion Coefficients for Non-Linear

Heat Conduction Problems

Samet Y. Kadioglu Robert R. Nourgaliev Vincent A. Mousseau

March 2008

Idaho National Laboratory Idaho Falls, Idaho 83415

Prepared for the U.S. Department of Energy

Through the INL LDRD Program Under DOE Idaho Operations Office

Contract DE-AC07-05ID14517

A Comparative Study of TheHarmonic and Arithmetic Averaging

of Diffusion Coefficients forNon-linear Heat Conduction

Problems

March 11, 2008

Samet Y. Kadioglu, Robert R. Nourgaliev, Vincent A. Mousseau,

Reactor Physics Analysis & Design, Idaho National Laboratory, P. O.Box 1625, MS 3840, Idaho Falls, ID 83415

1

Contents

1 Abstract 3

2 Introduction 4

3 Model equation 5

4 Numerical Algorithm 54.1 The Harmonic Average Model . . . . . . . . . . . . . . . . . . 54.2 The Arithmetic Average Model . . . . . . . . . . . . . . . . . 74.3 Truncation error analysis . . . . . . . . . . . . . . . . . . . . . 9

5 Numerical results 115.1 Moving-front Single-material test . . . . . . . . . . . . . . . . 115.2 Stationary-interface Multi-material test . . . . . . . . . . . . . 13

6 Conclusion 14

2

1 Abstract

We perform a comparative study for the harmonic versus arithmetic averag-ing of the heat conduction coefficient when solving non-linear heat transferproblems. In literature, the harmonic average is the method of choice, be-cause it is widely believed that the harmonic average is more accurate model.However, our analysis reveals that this is not necessarily true. For instance,we show a case in which the harmonic average is less accurate when a coarsermesh is used. More importantly, we demonstrated that if the boundary layersare finely resolved, then the harmonic and arithmetic averaging techniquesare identical in the truncation error sense. Our analysis further reveals thatthe accuracy of these two techniques depends on how the physical problemis modeled.

3

2 Introduction

In this paper, we study how the choice of averaging of the heat conductioncoefficient affects the solutions of non-linear heat conduction problems. Inliterature, the harmonic average is widely used by the heat transfer commu-nity. The harmonic average was promoted by Patankar [7]. It is designed tosolve heat transfer problems involving multiple material properties. Patankar[7] concluded that the harmonic average is more accurate in his case studies.Since then, the harmonic average became people’s choice when dealing withnon-linear heat conduction problems. To our knowledge, the harmonic aver-age is always assumed to be more accurate model regardless of the problemtypes, therefore no comparative study has been performed on this matter.

In this paper, we compare the harmonic average model with the arithmeticaverage model. We provide in-depth mathematical analysis and numericalcomputations. Our analysis and numerical results reveal that both modelsare identical for a very fine mesh. However, for coarser grids, in some casesthe harmonic average is better, in others the arithmetic average is better.For instance, in a single material test case, the harmonic average for coarsergrids performs very poorly to predict the thermal wave front location. Onthe other hand, the harmonic average (again for coarser grids) is found to bemore accurate when dealing with multi-material test cases. In Section 4, weprovide a detail mathematical analysis about the accuracy of the two models.Our analysis shows that the averaging techniques and the model problemsare tightly related. In other words, the model problem and the choice ofaveraging will directly affect the accuracy. Therefore, one should always beaware of the suitability of the averaging technique to the model problemhe/she is dealing with. A different way of stating this is that the averagingtechnique is a sub-grid physics model. The sub-grid physics models are basedon different assumptions. If these assumptions are matched in solving theproblem, then the model will be more accurate.

The organization of the present paper is as follows. In Section 3, we describethe partial differential equations of our model. In Section 4, we describe thenumerical algorithm employed and provide several mathematical derivationsand analysis. In Section 5, we present numerical results involving single andmulti-material test cases. Finally, Section 6 contains concluding remarks.

4

3 Model equation

We consider the following model equation,

∂T

∂t+

∂

∂x(D

∂T

∂x) = 0, (1)

representing a non-linear heat conduction problem. Where D is the non-linear heat conduction coefficient. In our particular case, we define D asD = T a. Here the constant a determines the strength of the non-linearity.(Note that setting a = 0 turns (1) into the linear diffusion equation). Thevariation in the heat conduction coefficient creates a fast moving wave front.This phenomena is also known as the Marshak wave ([5],[3]).

4 Numerical Algorithm

Our numerical algorithm is second order in both space and time, i.e, it isbased on the Crank-Nicolson scheme [9]. Our space discretization is doneconservatively based on a second order finite volume technique ([4],[10]).Our time integration is based on the Jacobien-Free Newton Krylov method[2].

The time and space discretization of Eqn. (1) can be written as:

T n+1i − T n

i

Δt+

Dn+1i+1/2(Tx)

n+1i+1/2 − Dn+1

i−1/2(Tx)n+1i−1/2

Δx= 0 (2)

where

(Tx)n+1i+1/2 =

T n+1i+1 − T n+1

i

Δx. (3)

The conductivity terms Di+1/2 and Di−1/2 are computed according to theharmonic or arithmetic averaging models.

4.1 The Harmonic Average Model

In this section, we show the derivation of the harmonic averaging of the heatconduction coefficient. Our derivation is based on two assumptions:

5

Cell edgesCell center

Di+1Di+1/2Di

Discontinuity at the cell edge

Half control volume

Figure 1: Conductivity representation for the harmonic average.

1) We assume the heat conduction coefficient is piecewise constant and con-tinuous from cell edge to cell edge(refer to Figure 1). 2) We consider steady-state solutions.

We know that the flux-in is equal to the flux-out at the (i + 1/2) face. Fromour first assumption, we also know that the conductivity terms (e.g., D’s)are equal from the cell center (i + 1) to the cell edge (i + 1/2), and from thecell edge (i + 1/2) to the (i)th cell (refer to Figure 1). Then we can write

Di+1

Ti+1 − Ti+1/2

Δx/2= Di

Ti+1/2 − Ti

Δx/2. (4)

Solving Eqn. (4) for Ti+1/2, we get

Ti+1/2 =Di+1Ti+1 + DiTi

Di+1 + Di. (5)

Now, we consider the following conservative discretizations;

T n+1i+1/4 − T n

i+1/4

Δt− Di+1/2Tx,i+1/2 − DiTx,i

Δx/2= 0, (6)

T n+1i+1/2 − T n

i+1/2

Δt− Di+1Tx,i+1 − DiTx,i

Δx= 0. (7)

Here, Eqn. (6) corresponds to the conservation law when considering the lefthalf part region between the (i)th and (i+1)th cells as the control volume(e.g.,Figure 1). Eqn. (7) corresponds to the conservation law when considering

6

the whole region between the (i)th and (i + 1)th cells as the control volume.At the steady state, the time part of Eqn. (6) and Eqn. (7) vanishes. Then,we can write

DiTx,i = Di+1Tx,i+1 = Di+1/2Tx,i+1/2. (8)

Now, considering the discretization of Eqn (8) in half and whole controlvolumes using forward differencing, we have

Di+1Ti+1 − Ti

Δx= Di+1/2

Ti+1/2 − Ti

Δx/2. (9)

Using this result and assumption 1), we arrive at

Di+1/2Ti+1 − Ti

Δx= Di

Ti+1/2 − Ti

Δx/2. (10)

Substituting Eqn. (5) in Eqn.(10) and performing the necessary algebra weobtain

Di+1/2 =2DiDi+1

Di + Di+1

. (11)

Eqn. (11) is called the harmonic average of the conductivity term at the(i + 1/2)th face.

4.2 The Arithmetic Average Model

The arithmetic average is derived based on the assumption that the con-ductivity term is piecewise constant and continuous from cell center to cellcenter (refer to Figure 2). This assumption and knowing that the flux-in isequal to the flux-out at the (i + 1/2) face give

Di+1/2

Ti+1 − Ti+1/2

Δx/2= Di+1/2

Ti+1/2 − Ti

Δx/2. (12)

Solving Eqn. (12) for Ti+1/2, we get

Ti+1/2 =Ti+1 + Ti

2. (13)

7

Cell edgesCell center

Di+1Di+1/2Di

Discontinuity at the cell center

Half control volume

Figure 2: Conductivity representation for the arithmetic average.

Now, making use of the steady state assumption and considering fluxes inthe left/right half and the whole control volumes (as we did in the harmoniccase), we have

Di+1Ti+1 − Ti

Δx= Di+1/2

Ti+1 − Ti+1/2

Δx/2, (14)

DiTi+1 − Ti

Δx= Di+1/2

Ti+1/2 − Ti

Δx/2. (15)

Adding Eqn. (14) and (15) from each side, we obtain

Di+1/2 =Di + Di+1

2. (16)

Eqn. (16) is called the arithmetic average of the conductivity term at the(i + 1/2)th face.

Remark-1: We remark that the assumptions we made in Sections 4.1 and4.2 are rather severe. For instance the constant conductivity term assumptionwould make sense for very fine grids. Furthermore, both models are derivedat the steady state. Therefore, during the transient, we should expect moreerrors associated with the model itself. We observed these behaviors in ournumerical results. For instance, in the single-material test case (Section 5.1),the harmonic model has more uncertainties for coarser grids leading more in-accuracies. On the other hand, in the multi-material test case, although theharmonic average is designed for this type of problem, during the transient,we don’t see significant advantages. However, at the steady state the har-

8

monic average is many order of magnitude better since all of the assumptionswe have made are matched (refer to Section 5.2).

Remark-2: Suppose, we have an insulator in the system, i.e, assume thei + 1/2th face corresponds to the boundary between the conductive materialon the left and the insulator on the right. Then, Di+1 → 0 in Eqn. (11),leading Di+1/2 → 0. This means zero heat flux at the i + 1/2th face asit should. However, Eqn. (16) gives none-zero heat flux as Di+1 → 0.Therefore, the harmonic average is the better choice (of course, one cansimply set the flux to zero at the insulator as a boundary condition), if weare interested in complete insulation.

Remark-3: In a radiation diffusion problem, if Ti > Ti+1, then Di >> Di+1,since typically D(T ) = T 3. With these, the harmonic average will favor thelowest conduction coefficient, then this will artificially damp the radiationand not allow it to flow into cold zones [6]. So in this case, the harmonicaverage is not the way to go. The arithmetic average or weighted arithmeticand geometric averages are suggested in [6]. Using the combination of theaveraging techniques is suggested in some other type of applications, e.g,shape optimization [1].

4.3 Truncation error analysis

In this section, we perform a truncation error analysis (refer to [9],[8]) for thetwo averaging techniques. The purpose of this analysis is to have a mathe-matical understanding about the errors committed by the two methods.

For the time being, we freeze the time discretization part of the PDE(i.e, ourinterest is the spatial error committed by the two models.). Then the PDEbecomes

∂

∂x(D

∂T

∂x) = 0. (17)

Discrete form of this equation can be written as;

1

Δx2[Di+1/2(Ti+1 − Ti) − Di−1/2(Ti − Ti−1)] = 0. (18)

The harmonic average:

9

The harmonic average considers continuous fluxes across the cell centers (Fig-ure 1), thus we perform Taylor series expansions centered at cell centers. Weconsider the following Taylor series expansions;

Ti+1 = Ti + ΔxTx,i +Δx2

2Txx,i +

Δx3

6Txxx,i +

Δx4

24Txxxx,i (19)

Ti−1 = Ti − ΔxTx,i +Δx2

2Txx,i − Δx3

6Txxx,i +

Δx4

24Txxxx,i (20)

Di+1/2 = Di +Δx

2Dx,i +

Δx2

8Dxx,i +

Δx3

48Dxxx,i +

Δx4

384Dxxxx,i (21)

Di−1/2 = Di − Δx

2Dx,i +

Δx2

8Dxx,i − Δx3

48Dxxx,i +

Δx4

384Dxxxx,i. (22)

Substituting these series expansions into Eqn. (18) ,canceling the commonterms with opposite signs, and making use of Eqn. (17), we obtain

τi =Δx2

24[Dx,iTxxx,i + DiTxxxx,i] + O(Δx4). (23)

This is the truncation error for the harmonic average.

The arithmetic average:

The arithmetic average considers continuous fluxes across the cell edges (Fig-ure 2), thus we expand Taylor series centered at cell edges. We consider thefollowing Taylor series expansions where we linearize at cell edges i + 1/2and i − 1/2;

Ti = Ti+1/2 − Δx

2Tx,i+1/2 +

Δx2

8Txx,i+1/2 − Δx3

48Txxx,i+1/2

+Δx4

384Txxxx,i+1/2 − Δx5

3840Txxxxx,i+1/2 (24)

Ti+1 = Ti+1/2 +Δx

2Tx,i+1/2 +

Δx2

8Txx,i+1/2 +

Δx3

48Txxx,i+1/2

+Δx4

384Txxxx,i+1/2 +

Δx5

3840Txxxxx,i+1/2 (25)

10

Ti = Ti−1/2 +Δx

2Tx,i−1/2 +

Δx2

8Txx,i−1/2 +

Δx3

48Txxx,i−1/2

+Δx4

384Txxxx,i−1/2 +

Δx5

3840Txxxxx,i−1/2 (26)

Ti−1 = Ti−1/2 − Δx

2Tx,i−1/2 +

Δx2

8Txx,i−1/2 − Δx3

48Txxx,i−1/2

+Δx4

384Txxxx,i−1/2 − Δx5

3840Txxxxx,i−1/2 (27)

Substituting these series expansions into Eqn. (18) , again canceling thecommon terms with opposite signs, and making use of Eqn. (17), we obtain

τi =Δx2

24

(Di+1/2Txxx,i+1/2 − Di−1/2Txxx,i−1/2)

Δx+ O(Δx4). (28)

This is the truncation error for the arithmetic average. Notice that whenΔx → 0 this expression (Eqn. (28)) is equivalent to

τi =Δx2

24[(DTxxx)x,i] + O(Δx4), (29)

or

τi =Δx2

24[Dx,iTxxx,i + DiTxxxx,i] + O(Δx4), (30)

which is identical to the truncation error for the harmonic average.

Note: This analysis clearly shows that when the mesh resolution is fineenough, the two procedures are identical. This finding will be verified nu-merically in the following section.

5 Numerical results

5.1 Moving-front Single-material test

We solve Eqn. (1) in [0, 1] with T (0, t) = 2.0, T (1, t) = 0.1, and T (x, 0) =0.1. With these settings, the nonlinear conductivity term creates a moving

11

wave front. The front locations are shown in Figures 3,5, and 8 for differentconductivity models and non-linearity.

Figures 3,5, and 8 show solutions at time = 0.08 with different mesh resolu-tions for a = 3, 2 and 1 respectively. As we can see from these figures, theharmonic average is very inaccurate for predicting the front location whena coarser mesh is used especially with larger a’s. Figures 4,6, and 9 showthe errors committed by the two methods again for a = 3, 2, 1. The errorsare calculated based on the comparison of the computed solution to a refer-ence solution (i.e, a reference solution for a = 3 is generated with 3200 gridpoints). Figures 4,6, and 9 indicate second order convergence (in space) withthe mesh refinements. Clearly, the harmonic average commits more errorfor coarser grids. On the other hand, for a very fine mesh the errors fromthe two methods are in the same order. This can be seen in Figure 7. Thisconclusion is also consistent with our truncation error analysis.

Another key observation from the error plots (Figures 4,6, and 9) is that theincrease in nonlinearity(i.e, a = 1, 2, 3) means more error in both methodsfor the same amount of grid points. This behavior can also be verified byconsidering our truncation error analysis. For instance, we can see this byconsidering the truncation terms, i.e, for the harmonic average for a = 1, 2, 3;

τa=1i =

Δx2

24[Txxx,i − TTxxxx,i] + O(Δx4), (31)

τa=2i =

Δx2

24[2TTxxx,i − T 2Txxxx,i] + O(Δx4), (32)

and

τa=3i =

Δx2

24[3T 2Txxx,i − T 3Txxxx,i] + O(Δx4), (33)

where D = T a and Dx = aT a−1 with a = 1, 2, 3. Then we obtain τa=2 =Tτa=1 + O(Δx2) and τa=3 = T 2τa=1 + O(Δx2).

We derive two conclusions from this test case. One is that the harmonicaverage is not better than the arithmetic average in contradiction to thecommon belief in literature. Second, both methods are convergent and com-parable with finer grids.

We also make the following observation regarding validation and uncertaintyquantification. Both harmonic and arithmetic averaging have been shown to

12

be second order accurate in space (Eqns. (23) and (28)). However, if thevalidation or uncertainty quantification are done at coarse grids, then theconclusions about uncertainty and accuracy would be very different. There-fore it is important to have a converged solution before the validation oruncertainty quantification is performed.

5.2 Stationary-interface Multi-material test

In this test, we solve the same PDE (e.g, Eqn. (1)), but we use constantin time (discontinuous in space, e.g, the discontinuity is located at the celledge) heat conduction coefficient, i.e,

D(x, t) =

{1000 if x ≤ 0.51 if x > 0.5

(34)

We solve this problem in the same domain and with the same boundaryconditions as in the single-material test case.

The discontinuity in the conductivity term represents a heat flow from a high-conductive material to a low-conductive material. We implemented bothaveraging techniques on this problem. We remark that a similar problem isstudied by Patankar [7] based on which it was suggested that the harmonicaverage is more accurate. Our results also favor the harmonic average forthis test when coarser grids are used. Nonetheless, with a fine-enough mesh,both methods are comparable.

Figure 10 shows the time history of the solution up to the steady state. Thetemperature builds up quickly in the high conductivity region(i.e, x ≤ 0.5),then slowly diffuses to the low conductive material. Figure 10 is producedusing 800 mesh points. We see that both methods give almost identicalsolutions. However the arithmetic average is slightly worse for coarser grids(i.e, Figure 11 with 20 points). This can be seen more clearly from Figure12 which compares the errors committed by the two methods at differenttimes. Notice that the error (especially for the arithmetic average) is higherat the front where the conductivity term changes dramatically. Also, Figure12 shows that errors for both methods first grow in time during the transient,and then decrease as the solution approaches steady state. This is because

13

we have both temporal and spatial errors added during the transient, yet thetemporal errors disappear at the steady state.

Figure 13 shows the log plots of the errors at different times. We observefrom Figure 13 that during the transient the errors from both methods arewithin an order of magnitude of each other. However, when the steady stateis reached and we have a diffusion coefficient that is constant from cell edgeto cell edge, both assumptions of the harmonic average sub-grid model aremet. In this case, the harmonic average is many magnitudes more accuratethan the arithmetic average.

Figure 14 shows convergence analysis. Clearly both methods are secondorder with the mesh refinement. Finally, Figure 15 tells us that the harmonicaverage is better for coarser grids, yet both methods are good for a fine grid.We remark that we had the opposite conclusion in the first test (i.e, thearithmetic average was better for coarser grids, refer to Figure 7).

6 Conclusion

We investigated the effect of the choice between the harmonic and arithmeticaveraging of the heat conduction coefficient on the heat transfer phenomena.We concluded that one should not favor one method over the other, becausewith a fine-enough mesh they are identical regardless of the model problem.However, we want to note that when using coarse grids one method can bebetter than the other depending on the model problem we are solving. Forinstance, in the single-material test, we found that the harmonic average isvery inaccurate (refer to Figure 7). On the other hand, the harmonic averageis more accurate in the multi-material test (refer to Figures 15 and 13). Itis important to note that many validation and uncertainty runs are done oncoarse grids. Thus, in this case, one has to be very careful when choosingthe averaging technique.

14

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

Tem

pera

ture

−−−−> x

Harmonic Averaging with a=3

M=800M=400M=200M=100M=50M=25

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Tem

pera

ture

−−−−> x

Arithmetic Averaging with a=3

M=800M=400M=200M=100M=50M=25

Figure 3: Comparison of the harmonic and arithmetic average for a = 3.Final run time = 8 × 10−2

−3.5 −3 −2.5 −2 −1.5 −1−5

−4

−3

−2

−1

0

1

log10

(Erro

r)

log10

(Δ x)

L2 norm of the error when a=3

Sec. OrderHarmonicArithmetic

Figure 4: Comparing L2 norm of the errors committed by the two methodsat time = 8 × 10−2.

15

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

Tem

pera

ture

−−−−> x


M=800M=400M=200M=100M=50M=25

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

Tem

pera

ture

−−−−> x


M=800M=400M=200M=100M=50M=25


−3 −2.5 −2 −1.5 −1−5

−4

−3

−2

−1

0

log10

(Erro

r)

log10

(Δ x)



Figure 6: Comparing L2 norm of the errors committed by the two methodsfor a = 2 and time = 8 × 10−2.

16

0 0.01 0.02 0.03 0.040

0.05

0.1

0.15

0.2

0.25

Erro

r

Δ x

L2 norm of the error with a=2

HarmonicArithmetic

Figure 7: L2 norm of the errors committed by the two methods for a = 2and time = 8 × 10−2. No log scale is used.

17

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

Tem

pera

ture

−−−−> x


M=800M=400M=200M=100M=50M=25

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

Tem

pera

ture

−−−−> x


M=800M=400M=200M=100M=50M=25


−2.8 −2.6 −2.4 −2.2 −2 −1.8 −1.6 −1.4−6

−5

−4

−3

−2

−1

log10

(Erro

r)

log10

(Δ x)



Figure 9: Comparing L2 norm of the errors committed by the two methodsfor a = 1 and time = 8 × 10−2.

18

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

Tem

pera

ture

−−−−> x

Harmonic Averaging with M=800 points.

t=1.0e−5

t=5.0e−5

t=1.0e−4

t=1.0e−3

t=1.0e−2

t=1.0e−1

t=1.0

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

Tem

pera

ture

−−−−> x

Arithmetic Averaging with M=800 points.

t=1.0e−5

t=5.0e−5

t=1.0e−4

t=1.0e−3

t=1.0e−2

t=1.0e−1

t=1.0

Figure 10: Time history of the solution by the two methods to the problem5.2. The solution reaches the steady state at time ≈ 1.0.

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

Tem

pera

ture

−−−−> x

Harmonic versus Arithmetic using M=20 points. Time=1.0e−4

HarmonicArithmeticReference Sol.

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

Tem

pera

ture

−−−−> x

Harmonic versus Arithmetic using M=20 points. Time=1.0

HarmonicArithmeticReference Sol.

Figure 11: Comparing coarse runs from the two averaging against the ref-erence solution. The left figure corresponds a transient solution, the rightfigure corresponds the steady state solution.

19

0 0.2 0.4 0.6 0.8 1−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

T−

Tre

f

−−−−> x

Harmonic Average

t=5.0e−5

t=1.0e−4

t=1.0e−3

t=1.0

0 0.2 0.4 0.6 0.8 1−0.2

0

0.2

0.4

0.6

0.8

T−

Tre

f

−−−−> x

Arithmetic Average

t=5.0e−5

t=1.0e−4

t=1.0e−3

t=1.0

Figure 12: Errors at different times by the two methods when comparing thecoarse (M = 20 grid points ) solution to the fine (M = 800 grid points )solution.

20

0 0.2 0.4 0.6 0.8 1−20

−15

−10

−5

0

log

10(T−

Tre

f )

−−−−> x

Error at time = 5.0e−5

ArithmeticHarmonic

0 0.2 0.4 0.6 0.8 1−20

−15

−10

−5

0

log

10(T−

Tre

f )

−−−−> x


ArithmeticHarmonic

0 0.2 0.4 0.6 0.8 1−12

−10

−8

−6

−4

−2

0

log

10(T−

Tre

f )

−−−−> x


ArithmeticHarmonic

0 0.2 0.4 0.6 0.8 1−12

−10

−8

−6

−4

−2

0

log

10(T−

Tre

f )

−−−−> x

Error at time = 1.0

ArithmeticHarmonic

Figure 13: Errors by the two methods when comparing the coarse (M = 20grid points ) solution to the fine (M = 800 grid points ) solution. Log scaleis used in the y-coordinate direction.

21

−3 −2.5 −2 −1.5 −1−4

−3

−2

−1

0

1

log10

(Erro

r)

log10

(Δ x)

L2 norm of the error for the multi−material test at time=1.0e−4


Figure 14: Convergence plots for the harmonic and arithmetic average whensolving the multi-material test problem.

0 0.02 0.04 0.06 0.08 0.10

0.05

0.1

0.15

0.2

Erro

r

Δ x

L2 norm of the error at time = 1.0e−4

HarmonicArithmetic

Figure 15: L2 norm of the errors when solving the multi-material test problemat time = 1 × 10−4. No log scale is used.

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References

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a comparative study of the harmonic and arithmetic

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